To arrange soldiers in the form of a square, the total number of soldiers must be a perfect square. When soldiers are left out, it means the given number of soldiers is not a perfect square. We need to find the largest perfect square less than the total number of soldiers and then subtract this perfect square from the total to find the number of soldiers left out.
Identify the total number of soldiers: The Army General has 40125 soldiers.
Understand the requirement: The soldiers are to be arranged in a square formation. This means the number of soldiers in the formation must be a perfect square (e.g., \(n \times n\)).
Find the largest perfect square less than 40125: We need to find the square root of 40125 and then consider the integer part of the square root.
Let's estimate the square root:
Since \(200^2 = 40000\) is less than 40125, and \(201^2 = 40401\) is greater than 40125, the largest perfect square less than 40125 is \(200^2\).
Calculate the number of soldiers in the square formation: The maximum number of soldiers that can be arranged in a square is \(200 \times 200 = 40000\).
Calculate the number of soldiers left out: Subtract the number of soldiers in the square formation from the total number of soldiers.
\[ \text{Soldiers left out} = \text{Total soldiers} - \text{Soldiers in square formation} \]\[ \text{Soldiers left out} = 40125 - 40000 \]\[ \text{Soldiers left out} = 125 \]B) 125 is the number of soldiers left out after forming the largest possible square.
Simplification of \(\frac{0.2\times0.2+0.02\times0.02-0.4\times0.02}{0.36}\) gives:
The problem requires simplifying a given expression involving decimals. We can recognize a common algebraic identity in the numerator, specifically \((a-b)^2 = a^2 - 2ab + b^2\), to simplify the calculation.
The given expression is:
\[ \frac{0.2\times0.2+0.02\times0.02-0.4\times0.02}{0.36} \]Let \(a = 0.2\) and \(b = 0.02\). Then, the numerator can be written as:
So, the numerator is \(a^2 + b^2 - 2ab\), which is the expansion of \((a-b)^2\).
Using \(a = 0.2\) and \(b = 0.02\), the numerator becomes:
\[ (0.2 - 0.02)^2 \]The denominator is \(0.36\). We can also write this as \((0.6)^2\).
To make the division easier, we can multiply the numerator and denominator by 10000 to remove the decimals:
\[ \frac{0.0324 \times 10000}{0.36 \times 10000} = \frac{324}{3600} \]Divide both numerator and denominator by common factors. Both are divisible by 36:
\[ \frac{324 \div 36}{3600 \div 36} = \frac{9}{100} = 0.09 \]B) 0.09 is the correct result after simplifying the given expression.
The question asks us to identify the true statement(s) among three given statements related to number systems. We need to recall the definitions of natural numbers, whole numbers, integers, and rational numbers and their relationships.
Based on the analysis, only statement 2 is true.
B) 2 — Statement 2 is true because every integer can be expressed as a fraction with a denominator of 1, thus satisfying the definition of a rational number.
The problem asks for the value of a trigonometric expression involving inverse trigonometric functions. We need to use a fundamental identity relating \(\sec^{-1}x\) and \(\operatorname{cosec}^{-1}x\).
This identity holds true for \(|x| \geq 1\), which is given in the problem statement.
A) 0 is the correct answer because, using the identity \[\sec^{-1}x + \operatorname{cosec}^{-1}x = \frac{\pi}{2}\], the expression simplifies to \(\cos\left(\frac{\pi}{2}\right)\), which is 0.
The question asks about the composition of the United Nations Security Council (UNSC) as of October 2020. This requires knowledge of the structure and membership categories of the UNSC.
Correct Option: A) 5 permanent members and 10 non-permanent members
To solve this blood relation problem, we will establish the relationships step-by-step, building a family tree or a chain of relations from the given statements. We will then deduce the relationship between X and A.
Correct Option: B) Daughter's Daughter
The problem asks us to find half of a number, given a relationship involving fractions of that number. We will first set up an equation based on the given information to find the number, and then calculate half of that number.
Correct Option: A) 175 is half of the number 350.
We need to determine a five-digit number based on given clues about its digits. We will assign variables to each digit's place value and then use the given conditions to find the value of each digit sequentially.
A five-digit number can be represented as \(ABCDE\), where:
Digit in the hundred's place: The problem states that the digit in the hundred's place is 2.
\[ C = 2 \]Digit in the unit's place: The digit in the unit's place is twice the digit in the hundred's place.
\[ E = 2 \times C = 2 \times 2 = 4 \]Digit in the thousand's place: The number has no thousands.
\[ B = 0 \]Digit in the ten thousand's place: The digit in the ten thousand's place is the sum of the digit in the hundred's place and the digit in the unit's place.
\[ A = C + E = 2 + 4 = 6 \]Digit in the ten's place: The digit in the ten's place is the digit in the ten thousand's place minus 1.
\[ D = A - 1 = 6 - 1 = 5 \]Form the number: Now, assemble the digits in their respective places:
The number is \(60254\).
Correct Option: A) 60254 is the number derived from following all the given conditions.
A is 50% as efficient as B. C completes half of the task completed by A and B together in a day. If C alone completes the task in 40 days, then A, B and C together can complete the task in:
13 \( \frac{1}{3} \) days
13 days
\( \frac{43}{3} \) days
\( \frac{1}{3} \) days
This problem involves work and time. We need to find the combined work rate of A, B, and C to determine the time they take to complete the task together. We will use the given efficiencies and individual work times to calculate the daily work done by each person.
Determine C's daily work:
C completes the task in 40 days. So, C's one day's work is:
\[ \text{C's 1 day's work} = \frac{1}{40} \]Relate C's work to A and B's work:
The problem states that C completes half of the task completed by A and B together in a day. Let \((A+B)\)'s one day's work be \(W_{A+B}\).
\[ \text{C's 1 day's work} = \frac{1}{2} \times W_{A+B} \]\[ \frac{1}{40} = \frac{1}{2} \times W_{A+B} \]\[ W_{A+B} = \frac{2}{40} = \frac{1}{20} \]So, A and B together complete the task in 20 days. This means A's one day's work + B's one day's work \( = \frac{1}{20} \).Determine A's and B's individual daily work:
A is 50% as efficient as B. This means if B does \(x\) units of work in a day, A does \(0.5x\) units of work in a day. If B's one day's work is \(\frac{1}{B_{\text{days}}}\), then A's one day's work is \(\frac{1}{A_{\text{days}}}\).
Let B's one day's work be \(W_B\). Then A's one day's work is \(W_A = 0.5 \times W_B\).
We know \(W_A + W_B = \frac{1}{20}\).
Substitute \(W_A\):
\[ 0.5 W_B + W_B = \frac{1}{20} \]\[ 1.5 W_B = \frac{1}{20} \]\[ W_B = \frac{1}{20 \times 1.5} = \frac{1}{30} \]So, B's one day's work is \(\frac{1}{30}\). This means B completes the task in 30 days.Now find A's one day's work:
\[ W_A = 0.5 \times W_B = 0.5 \times \frac{1}{30} = \frac{1}{60} \]So, A's one day's work is \(\frac{1}{60}\). This means A completes the task in 60 days.Calculate the combined daily work of A, B, and C:
Combined one day's work \((W_{A+B+C})\) is the sum of their individual daily works:
\[ W_{A+B+C} = W_A + W_B + W_C \]\[ W_{A+B+C} = \frac{1}{60} + \frac{1}{30} + \frac{1}{40} \]Find a common denominator, which is 120:
\[ W_{A+B+C} = \frac{2}{120} + \frac{4}{120} + \frac{3}{120} \]\[ W_{A+B+C} = \frac{2 + 4 + 3}{120} = \frac{9}{120} \]\[ W_{A+B+C} = \frac{3}{40} \]Calculate the total time taken by A, B, and C together:
If their combined one day's work is \(\frac{3}{40}\) of the task, then they will complete the task in the reciprocal of this fraction:
\[ \text{Time} = \frac{1}{W_{A+B+C}} = \frac{1}{\frac{3}{40}} = \frac{40}{3} \text{ days} \]Convert this to a mixed fraction:\[ \frac{40}{3} = 13 \frac{1}{3} \text{ days} \]A) \(13 \frac{1}{3}\) days is the correct answer, as calculated above.
To find the time required to cover a certain distance at a uniform speed, we use the fundamental formula relating distance, speed, and time. We will first calculate the time in hours and then convert the fractional part of an hour into minutes.
The relationship between distance, speed, and time is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]First, express \(\frac{10}{3}\) as a mixed number:
\[ \frac{10}{3}\ \text{h} = 3\ \frac{1}{3}\ \text{h} \]Now, convert the fractional part (\(\frac{1}{3}\) h) into minutes:
\[ \frac{1}{3}\ \text{h} = \frac{1}{3} \times 60\ \text{minutes} = 20\ \text{minutes} \]So, the total time required is 3 hours and 20 minutes.
Correct Option: D) 3 h 20 min